Talk:Final topology

Latest comment: 6 years ago by 136.159.16.20 in topic Cone versus Cocone

Coarsest topology

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I'm removing the following from the article:

  • Given a family of topologies {τi} on a fixed set X the final topology on X with respect to the functions idX : (X, τi) → X is the infimum (or meet) of the topologies {τi} in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies {τi}.

I beleive this one is supposed to describe the initial topology not the final topology: The intersection of two topologies will be coarser or weaker than any of the individual τi. Elsewhere, the article describes the final topology as being the "finest" not the "coarsest".

I'll be removing a similar paragraph from the initial topology article. linas 20:14, 20 November 2005 (UTC)Reply

Be careful Linas; the example is correct. You are confusing greatest element with greatest lower bound. The topology on X must be coarser than each given topology to ensure continuity (i.e. a lower bound), but it is the finest topology which is coarser than each given one (i.e. a greatest lower bound). -- Fropuff 16:15, 21 November 2005 (UTC)Reply
Right. Thanks. Sorry. linas 20:27, 25 November 2006 (UTC)Reply

Examples

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The article needs an example of a topology that satisfies conditions form being an inital toplogy, but is not a final topology. linas 20:14, 20 November 2005 (UTC)Reply

Projective vs. inductive topology

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I'm quite sure that the final topology is also called "inductive topology" and the initial topology is the "projective topology", thus there is a serious mistake in the first sentence (and in the first sentence of the article about the initial topology). This would also be consistent with "inductive topology" redirecting to "final topology" and "projective topology" redirecting to "initial topology" - so either the redirections are wrong or the articles. I think this should be fixed asap, but I haven't changed anything yet because I don't have a reference at hand. TSBM (talk) — Preceding undated comment added 12:00, 8 December 2014 (UTC)Reply

On the given universal property

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"The final topology on   can be characterized by the following universal property: a function   from   to some space   is continuous if and only if   is continuous for each iI."

Well, not really. This universal property is for the topological space  . The universal property of   is the one given at the beginning:

  is the finest topology such that each

 

is continuous.

This means explicitly: For every topology   on   such that the   are continuous,  . (The inclusion   can be viewed as a morphism in the category of topologies on  .)

Claus from Leipzig (talk) 12:25, 17 October 2017 (UTC)Reply

Cone versus Cocone

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In the categorical description section, cones are used when cocones ( F(X) \to N ) might be desired? — Preceding unsigned comment added by 136.159.16.20 (talk) 03:57, 24 February 2018 (UTC)Reply